Sysquake Pro – Table of Contents
Sysquake for LaTeX – Table of Contents
Array Functions
arrayfun
Function evaluation for each element of an array.
Syntax
(B1, ...) = arrayfun(fun, A1, ...)
Description
arrayfun(fun,A) evaluates function fun for each element of numeric array A. Each evaluation must give a scalar result of numeric (or logical or char) type; results are returned as a numeric array the same size as A. First argument is a function reference, an inline function, or the name of a function as a string.
With more than two input arguments, arrayfun calls function fun as feval(fun,A1(i),A2(i),...). All array arguments must have the same size, but their type can be different.
With two output arguments or more, arrayfun evaluates function fun with the same number of output arguments and builds a separate array for each output. Without output argument, arrayfun evaluates fun without output argument.
arrayfun differs from cellfun: all input arguments of arrayfun are arrays of any type (not necessarily cell arrays), and corresponding elements are provided provided to fun. With map, input arguments as well as output arguments are cell arrays.
Examples
arrayfun(@isempty, {1, ''; {}, ones(5)})
F T
T F
map(@isempty, {1, ''; {}, ones(5)})
2x2 cell array
(m, n) = arrayfun(@size, {1, ''; {}, ones(2, 5)})
m =
1 0
0 2
n =
1 0
0 5
See also
cat
Array concatenation.
Syntax
cat(dim, A1, A2, ...)
Description
cat(dim,A1,A2,...) concatenates arrays A1, A2, etc. along dimension dim. Other dimensions must match. cat is a generalization of the comma and the semicolon inside brackets.
Examples
cat(2, [1,2;3,4], [5,6;7,8])
1 2 5 6
3 4 7 8
cat(3, [1,2;3,4], [5,6;7,8])
2x2x2 array
(:,:,1) =
1 2
3 4
(:,:,2) =
5 6
7 8
See also
operator [], operator ;, operator ,
cell
Cell array of empty arrays.
Syntax
C = cell(n) C = cell(n1,n2,...) C = cell([n1,n2,...])
Description
cell builds a cell array whose elements are empty arrays []. The size of the cell array is specified by one integer for a square array, or several integers (either as separate arguments or in a vector) for a cell array of any size.
Example
cell(2, 3) 2x3 cell array
See also
cellfun
Function evaluation for each cell of a cell array.
Syntax
A = cellfun(fun, C) A = cellfun(fun, C, ...) A = cellfun(fun, S) A = cellfun(fun, S, ...)
Description
cellfun(fun,C) evaluates function fun for each cell of cell array C. Each evaluation must give a scalar result of numeric, logical, or character type; results are returned as a non-cell array the same size as C. First argument is a function reference, an inline function, or the name of a function as a string.
With more than two input arguments, cellfun calls function fun as feval(fun,C{i},other), where C{i} is each cell of C in turn, and other stands for the remaining arguments of cellfun.
The second argument can be a structure array S instead of a cell array. In that case, fun is called with S(i).
cellfun differs from map in two ways: the result is a non-cell array, and remaining arguments of cellfun are provided directly to fun.
Examples
cellfun(@isempty, {1, ''; {}, ones(5)})
F T
T F
map(@isempty, {1, ''; {}, ones(5)})
2x2 cell array
cellfun(@size, {1, ''; {}, ones(5)}, 2)
1 0
0 5
See also
diag
Creation of a diagonal matrix or extraction of the diagonal elements of a matrix.
Syntax
M = diag(v) M = diag(v,k) v = diag(M) v = diag(M,k)
Description
With a vector input argument, diag(v) creates a square diagonal matrix whose main diagonal is given by v. With a second argument, the diagonal is moved by that amount in the upper right direction for positive values, and in the lower left direction for negative values.
With a matrix input argument, the main diagonal is extracted and returned as a column vector. A second argument can be used to specify another diagonal.
Examples
diag(1:3) 1 0 0 0 2 0 0 0 3 diag(1:3,1) 0 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 M = magic(3) M = 8 1 6 3 5 7 4 9 2 diag(M) 8 5 2 diag(M,1) 1 7
See also
eye
Identity matrix.
Syntax
M = eye(n) M = eye(m,n) M = eye([m,n]) M = eye(..., type)
Description
eye builds a matrix whose diagonal elements are 1 and other elements 0. The size of the matrix is specified by one integer for a square matrix, or two integers (either as two arguments or in a vector of two elements) for a rectangular matrix.
An additional input argument can be used to specify the type of the result. It must be the string 'double', 'single', 'int8', 'int16', 'int32', 'int64', 'uint8', 'uint16', 'uint32', or 'uint64' (64-bit arrays are not supported on all platforms).
Examples
eye(3)
1 0 0
0 1 0
0 0 1
eye(2, 3)
1 0 0
0 1 0
eye(2, 'int8')
2x2 int8 array
1 0
0 1
See also
fevalx
Function evaluation with array expansion.
Syntax
(Y1,...) = fevalx(fun,X1,...)
Description
(Y1,Y2,...)=fevalx(fun,X1,X2,...) evaluates function fun with input arguments X1, X2, etc. Arguments must be arrays, which are expanded if necessary along singleton dimensions so that all dimensions match. For instance, three arguments of size 3x1x2, 1x5 and 1x1 are replicated into arrays of size 3x5x2. Output arguments are assigned to Y1, Y2, etc. Function fun is specified either by its name as a string, by a function reference, or by an inline or anonymous function.
Example
fevalx(@plus, 1:5, (10:10:30)')
11 12 13 14 15
21 22 23 24 25
31 32 33 34 35
See also
feval, meshgrid, repmat, inline, operator @
find
Find the indices of the non-null elements of an array.
Syntax
ix = find(v) [s1,s2] = find(M) [s1,s2,x] = find(M) ... = find(..., n) ... = find(..., n, dir)
Description
With one output argument, find(v) returns a vector containing the indices of the nonzero elements of v. v can be an array of any dimension; the indices correspond to the internal storage ordering and can be used to access the elements with a single subscript.
With two output arguments, find(M) returns two vectors containing the subscripts (row in the first output argument, column in the second output argument) of the nonzero elements of 2-dim array M. To obtain subscripts for an array of higher dimension, you can convert the single output argument of find to subscripts with ind2sub.
With three output arguments, find(M) returns in addition the nonzero values themselves in the third output argument.
With a second input argument n, find limits the maximum number of elements found. It searches forward by default; with a third input argument dir, find gives the n first nonzero values if dir is 'first' or 'f', and the n last nonzero values if dir is 'last' or 'l'.
Examples
ix = find([1.2,0;0,3.6])
ix =
1
4
[s1,s2] = find([1.2,0;0,3.6])
s1 =
1
2
s2 =
1
2
[s1,s2,x] = find([1.2,0;0,3.6])
s1 =
1
2
s2 =
1
2
x =
1.2
3.6
A = rand(3)
A =
0.5599 0.3074 0.5275
0.3309 0.8077 0.3666
0.7981 0.6424 0.6023
find(A > 0.7, 2, 'last')
7
5
See also
flipdim
Flip an array along any dimension.
Syntax
B = flipdim(A, dim)
Description
flipdim(A,dim) gives an array which has the same size as A, but where indices of dimension dim are reversed.
Examples
flipdim(cat(3, [1,2;3,4], [5,6;7,8]), 3)
2x2x2 array
(:,:,1) =
5 6
7 8
(:,:,2) =
1 2
3 4
See also
fliplr, flipud, rot90, reshape
fliplr
Flip an array or a list around its vertical axis.
Syntax
A2 = fliplr(A1) list2 = fliplr(list1)
Description
fliplr(A1) gives an array A2 which has the same size as A1, but where all columns are placed in reverse order.
fliplr(list1) gives a list list2 which has the same length as list1, but where all top-level elements are placed in reverse order. Elements themselves are left unchanged.
Examples
fliplr([1,2;3,4])
2 1
4 3
fliplr({1, 'x', {1,2,3}})
{{1,2,3}, 'x', 1}
See also
flipud, flipdim, rot90, reshape
flipud
Flip an array upside-down.
Syntax
A2 = flipud(A1)
Description
flipud(A1) gives an array A2 which has the same size as A1, but where all lines are placed in reverse order.
Example
flipud([1,2;3,4]) 3 4 1 2
See also
fliplr, flipdim, rot90, reshape
ind2sub
Conversion from single index to row/column subscripts.
Syntax
(i, j, ...) = ind2sub(size, ind)
Description
ind2sub(size,ind) gives the subscripts of the element which would be retrieved from an array whose size is specified by size by the single index ind. size must be either a scalar for square matrices or a vector of two elements or more for arrays. ind can be an array; the result is calculated separately for each element and has the same size.
Example
M = [3, 6; 8, 9];
M(3)
8
(i, j) = ind2sub(size(M), 3)
i =
2
j =
1
M(i, j)
8
See also
interp1
1D interpolation.
Syntax
yi = interp1(x, y, xi) yi = interp1(x, y, xi, method) yi = interp1(y, xi) yi = interp1(y, xi, method) yi = interp1(..., method, extraval)
Description
interp1(x,y,xi) interpolates data along one dimension. Input data are defined by vector y, where element y(i) corresponds to coordinates x(i). Interpolation is performed at points defined in vector xi; the result is a vector of the same size as xi.
If y is an array, interpolation is performed along dimension 1 (i.e. along its columns), and size(y,1) must be equal to length(x). Then if xi is a vector, interpolation is performed at the same points for each remaining dimensions of y, and the result is an array of size [length(xi),size(y)(2:end)]; if xi is an array, all sizes must match y except for the first one.
If x is missing, it defaults to 1:size(y,1).
The default interpolation method is piecewise linear. An additional input argument can be provided to specify it with a string (only the first character is considered):
| Argument | Meaning |
|---|---|
| '0' or 'nearest' | nearest value |
| '<' | lower coordinate |
| '>' | higher coordinate |
| '1' or 'linear' | piecewise linear |
| '3' or 'cubic' | piecewise cubic |
| 'p' or 'pchip' | pchip |
Cubic interpolation gives continuous values and first derivatives, and null second derivatives at end points. Pchip (piecewise cubic Hermite interpolation) also gives continuous values and first derivatives, but guarantees that the interpolant stays within the limits of the data in each interval (in particular monotonicity is preserved) at the cost of larger second derivatives.
With vectors, interp1 produces the same result as interpn; vector orientations do not have to match, though.
When the method is followed by a scalar number extraval, that value is assigned to all values outside the range defined by x (i.e. extrapolated values). The default is NaN.
Examples
One-dimension interpolation:
interp1([1, 2, 5, 8], [0.1, 0.2, 0.5, 1], [0, 2, 3, 7]) nan 0.2000 0.3000 0.8333 interp1([1, 2, 5, 8], [0.1, 0.2, 0.5, 1], [0, 2, 3, 7], '0') nan 0.2000 0.2000 1.0000
Interpolation of multiple values:
t = 0:10;
y = [sin(t'), cos(t')];
tnew = 0:0.4:8;
ynew = interp1(t, y, tnew)
ynew =
0.0000 1.0000
0.3366 0.8161
...
0.8564 0.2143
0.9894 -0.1455
See also
interpn
Multidimensional interpolation.
Syntax
Vi = interpn(x1, ..., xn, V, xi1, ..., xin) Vi = interpn(x1, ..., xn, V, xi1, ..., xin, method) Vi = interpn(..., method, extraval)
Description
interpn(x1,...,xn,V,xi1,...,xin) interpolates data in a space of n dimensions. Input data are defined by array V, where element V(i,j,...) corresponds to coordinates x1(i), x2(j), etc. Interpolation is performed for each coordinates defined by arrays xi1, xi2, etc., which must all have the same size; the result is an array of the same size.
Length of vectors x1, x2, ... must match the size of V along the corresponding dimension. Vectors x1, x2, ... must be sorted (monotonically increasing or decreasing), but they do not have to be spaced uniformly. Interpolated points outside the input volume are set to nan. Input and output data can be complex. Imaginary parts of coordinates are ignored.
The default interpolation method is multilinear. An additional input argument can be provided to specify it with a string (only the first character is considered):
| Argument | Meaning |
|---|---|
| '0' or 'nearest' | nearest value |
| '<' | lower coordinates |
| '>' | higher coordinates |
| '1' or 'linear' | multilinear |
Method '<' takes the sample where each coordinate has its index as large as possible, lower or equal to the interpolated value, and smaller than the last coordinate. Method '>' takes the sample where each coordinate has its index greater or equal to the interpolated value.
When the method is followed by a scalar number extraval, that value is assigned to all values outside the input volume (i.e. extrapolated values). The default is NaN.
Examples
One-dimension interpolation:
interpn([1, 2, 5, 8], [0.1, 0.2, 0.5, 1], [0, 2, 3, 7]) nan 0.2000 0.3000 0.8333 interpn([1, 2, 5, 8], [0.1, 0.2, 0.5, 1], [0, 2, 3, 7], '0') nan 0.2000 0.2000 1.0000
Three-dimension interpolation:
D = cat(3,[0,1;2,3],[4,5;6,7]); interpn([0,1], [0,1], [0,1], D, 0.2, 0.7, 0.5) 3.1000
Image rotation (we define original coordinates between -0.5 and 0.5 in vector c and arrays X and Y, and the image as a linear gradient between 0 and 1):
c = -0.5:0.01:0.5; X = repmat(c, 101, 1); Y = X'; phi = 0.2; Xi = cos(phi) * X - sin(phi) * Y; Yi = sin(phi) * X + cos(phi) * Y; D = 0.5 + X; E = interpn(c, c, D, Xi, Yi); E(isnan(E)) = 0.5;
See also
intersect
Set intersection.
Syntax
c = intersect(a, b) (c, ia, ib) = intersect(a, b)
Description
intersect(a,b) gives the intersection of sets a and b, i.e. it gives the set of members of both sets a and b. Sets are any type of numeric, character or logical arrays, or lists or cell arrays of character strings. Multiple elements of input arguments are considered as single members; the result is always sorted and has unique elements.
The optional second and third output arguments are vectors of indices such that if (c,ia,ib)=intersect(a,b), then c is a(ia) as well as b(ib).
Example
a = {'a','bc','bbb','de'};
b = {'z','bc','aa','bbb'};
(c, ia, ib) = intersect(a, b)
c =
{'bbb','bc'}
ia =
3 2
ib =
4 2
a(ia)
{'bbb','bc'}
b(ib)
{'bbb','bc'}
Set exclusive or can also be computed as the union of a and b minus the intersection of a and b:
setdiff(union(a, b), intersect(a, b))
{'a','aa','de','z'}
See also
unique, union, setdiff, setxor, ismember
inthist
Histogram of an integer array.
Syntax
h = inthist(A, n)
Description
inthist(A,n) computes the histogram of the elements of integer array A between 0 and n-1. A must have an integer type (int8/16/32/64 or uint8/16/32/64). The result is a row vector h of length n, where h(i) is the number of elements in A with value i-1.
Example
A = map2int(rand(100), 0, 1, 'uint8');
h = inthist(A, 10)
h =
37 31 34 34 32 35 38 36 36 32
See also
ipermute
Inverse permutation of the dimensions of an array.
Syntax
B = ipermute(A, perm)
Description
ipermute(A,perm) returns an array with the same elements as A, but where dimensions are permuted according to the vector of dimensions perm. It performs the inverse permutation of permute. perm must contain integers from 1 to n; dimension i in A becomes dimension perm(i) in the result.
Example
size(ipermute(rand(3,4,5), [2,3,1])) 5 3 4
See also
isempty
Test for empty array, list or struct.
Syntax
b = isempty(A) b = isempty(list) b = isempty(S)
Description
isempty(obj) gives true if obj is the empty array [] of any type (numeric, char, logical or cell array) or the empty struct, and false otherwise.
Examples
isempty([])
true
isempty(0)
false
isempty('')
true
isempty({})
true
isempty({{}})
false
isempty(struct)
true
See also
iscell
Test for cell arrays.
Syntax
b = iscell(X)
Description
iscell(X) gives true if X is a cell array or a list, and false otherwise.
Examples
iscell({1;2})
true
iscell({1,2})
true
islist({1;2})
false
See also
ismember
Test for set membership.
Syntax
b = ismember(m, s) (b, ix) = ismember(m, s)
Description
ismember(m,s) tests if elements of array m are members of set s. The result is a logical array the same size as m; each element is true if the corresponding element of m is a member of s, or false otherwise. m must be a numeric array or a cell array, matching type of set s.
With a second output argument ix, ismember also gives the index of the corresponding element of m in s, or 0 if the element is not a member of s.
Example
s = {'a','bc','bbb','de'};
m = {'d','a','x';'de','a','z'};
(b, ix) = ismember(m, s)
b =
F T F
T T F
ix =
0 1 0
4 1 0
See also
intersect, union, setdiff, setxor
length
Length of a vector or a list.
Syntax
n = length(v) n = length(list)
Description
length(v) gives the length of vector v. length(A) gives the number of elements along the largest dimension of array A. length(list) gives the number of elements in a list.
Examples
length(1:5)
5
length((1:5)')
5
length(ones(2,3))
3
length({1, 1:6, 'abc'})
3
length({{}})
1
See also
linspace
Sequence of linearly-spaced elements.
Syntax
v = linspace(x1, x2) v = linspace(x1, x2, n)
Description
linspace(x1,x2) produces a row vector of 50 values spaced linearly from x1 to x2 inclusive. With a third argument, linspace(x1,x2,n) produces a row vector of n values.
Examples
linspace(1,10) 1.0000 1.1837 1.3673 ... 9.8163 10.0000 linspace(1,2,6) 1.0 1.2 1.4 1.6 1.8 2.0
See also
logspace
Sequence of logarithmically-spaced elements.
Syntax
v = logspace(x1, x2) v = logspace(x1, x2, n)
Description
logspace(x1,x2) produces a row vector of 50 values spaced logarithmically from 10^x1 to 10^x2 inclusive. With a third argument, logspace(x1,x2,n) produces a row vector of n values.
Example
logspace(0,1) 1.0000 1.0481 1.0985 ... 9.1030 9.5410 10.0000
See also
magic
Magic square.
Syntax
M = magic(n)
Description
A magic square is a square array of size n-by-n which contains each integer
between 1 and
There is no 2-by-2 magic square. If the size is 2, the matrix [1,3;4,2] is returned instead.
Example
magic(3) 8 1 6 3 5 7 4 9 2
See also
meshgrid
Arrays of X-Y coordinates.
Syntax
(X, Y) = meshgrid(x, y) (X, Y) = meshgrid(x)
Description
meshgrid(x,y) produces two arrays of x and y coordinates suitable for the evaluation of a function of two variables. The input argument x is copied to the rows of the first output argument, and the input argument y is copied to the columns of the second output argument, so that both arrays have the same size. meshgrid(x) is equivalent to meshgrid(x,x).
Example
(X, Y) = meshgrid(1:5, 2:4)
X =
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
Y =
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4
Z = atan2(X, Y)
Z =
0.4636 0.7854 0.9828 1.1071 1.1903
0.3218 0.5880 0.7854 0.9273 1.0304
0.2450 0.4636 0.6435 0.7854 0.8961
See also
ndgrid
Arrays of N-dimension coordinates.
Syntax
(X1, ..., Xn) = ndgrid(x1, ..., xn) (X1, ..., Xn) = ndgrid(x)
Description
ndgrid(x1,...,xn) produces n arrays of n dimensions. Array i is obtained by reshaping input argument i as a vector along dimension i and replicating it along all other dimensions to match the length of other input vectors. All output arguments have the same size.
With one input argument, ndgrid reuses it to match the number of output arguments.
(Y,X)=ndgrid(y,x) is equivalent to (X,Y)=meshgrid(x,y).
Example
(X1, X2) = ndgrid(1:3)
X1 =
1 1 1
2 2 2
3 3 3
X2 =
1 2 3
1 2 3
1 2 3
See also
ndims
Number of dimensions of an array.
Syntax
n = ndims(A)
Description
ndims(A) returns the number of dimensions of array A, which is at least 2. Scalars, row and column vectors, and matrices have 2 dimensions.
Examples
ndims(magic(3)) 2 ndims(rand(3,4,5)) 3
See also
size, squeeze, permute, ipermute
nnz
Number of nonzero elements.
Syntax
n = nnz(A)
Description
nnz(A) returns the number of nonzero elements of array A. Argument A must be a numeric, char or logical array.
Examples
nnz(-2:2) 4 nnz(magic(3) > 3) 6
See also
num2cell
Conversion from numeric array to cell array.
Syntax
C = num2cell(A) C = num2cell(A, dims)
Description
num2cell(A) creates a cell array the same size as numeric array A. The value of each cell is the corresponding elements of A.
num2cell(A,dims) cuts array A along the dimensions not in dims and creates a cell array with the result. Dimensions of cell array are the same as dimensions of A for dimensions not in dims, and 1 for dimensions in dims; dimensions of cells are the same as dimensions of A for dimensions in dims, and 1 for dimensions not in dims.
Argument A can be a numeric array of any dimension and class, a logical array, or a char array.
Examples
num2cell([1, 2; 3, 4])
{1, 2; 3, 4}
num2cell([1, 2; 3, 4], 1)
{[1; 3], [2; 4]}
num2cell([1, 2; 3, 4], 2)
{[1, 2]; [3, 4]}
See also
numel
Number of elements of an array.
Syntax
n = numel(A)
Description
numel(A) gives the number of elements of array A. It is equivalent to prod(size(A)).
Examples
numel(1:5)
5
numel(ones(2, 3))
6
numel({1, 1:6; 'abc', []})
4
numel({2, 'vwxyz'})
2
See also
ones
Array of ones.
Syntax
A = ones(n) A = ones(n1, n2, ...) A = ones([n1, n2, ...]) A = ones(..., type)
Description
ones builds an array whose elements are 1. The size of the array is specified by one integer for a square matrix, or several integers (either as separate arguments or in a vector) for an array of any size.
An additional input argument can be used to specify the type of the result. It must be the string 'double', 'single', 'int8', 'int16', 'int32', 'int64', 'uint8', 'uint16', 'uint32', or 'uint64' (64-bit arrays are not supported on all platforms).
Examples
ones(2,3)
1 1 1
1 1 1
ones(2, 'int32')
2x2 int32 array
1 1
1 1
See also
zeros, eye, rand, randn, repmat
permute
Permutation of the dimensions of an array.
Syntax
B = permute(A, perm)
Description
permute(A,perm) returns an array with the same elements as A, but where dimensions are permuted according to the vector of dimensions perm. It is a generalization of the matrix transpose operator. perm must contain integers from 1 to n; dimension perm(i) in A becomes dimension i in the result.
Example
size(permute(rand(3,4,5), [2,3,1])) 4 5 3
See also
ndims, squeeze, ipermute, num2cell
rand
Uniformly-distributed random number.
Syntax
x = rand
A = rand(n)
A = rand(n1, n2, ...)
A = rand([n1, n2, ...])
A = rand(..., type)
rand('seed', s);
Description
rand builds a scalar pseudo-random number uniformly distributed between 0 and 1. The lower bound 0 may be reached, but the upper bound 1 is never. The default generator is based on a scalar 64-bit seed, which theoretically has a period of 2^64-2^32 numbers. This seed can be set with the arguments rand('seed',s), where s is a scalar. rand('seed',s) returns the empty array [] as output argument. To discard it, the statement should be followed by a semicolon. The generator can be changed with rng.
rand(n), rand(n1,n2,...) and rand([n1,n2,...]) return an n-by-n square array or an array of arbitrary size whose elements are pseudo-random numbers uniformly distributed between 0 and 1.
An additional input argument can be used to specify the type of the result, 'double' (default) or 'single'. With the special value 'raw', rand returns an unscaled integer result of type double which corresponds to the uniform output of the random generator before it is mapped to the range between 0 and 1. The scaling factor can be retrieved in the field rawmax of the structure returned by rng.
Examples
rand
0.2361
rand(1, 3)
0.6679 0.8195 0.2786
rand('seed',0);
rand
0.2361
See also
randi
Uniformly-distributed integer random number.
Syntax
x = randi(nmax) x = randi(range) M = randi(..., n) M = randi(..., n1, n2, ...) M = randi(..., [n1, n2, ...]) M = randi(..., class)
Description
randi(nmax) produces a scalar pseudo-random integer number uniformly distributed between 1 and nmax. randi(range), where range is a two-element vector [nmin,nmax], produces a scalar pseudo-random integer number uniformly distributed between nmin and nmax.
With more numeric input arguments, randi produces arrays of pseudo-random integer numbers. randi(range,n) produces an n-by-n square array, and randi(range,[n1,n2,...]) or randi(range,n1,n2,...) produces an array of the specified size.
The number class of the result can be specified with a final string argument. The default is 'double'.
Examples
randi(10)
3
randi(10, [1, 5])
3 4 6 8 1
randi([10,15], [1, 5])
12 14 13 10 13
randi(8, [1, 5], 'uint8')
1x5 uint8 array
3 4 5 7 2
See also
randn
Normally-distributed random number
Syntax
x = randn
A = randn(n)
A = randn(n1, n2, ...)
A = randn([n1, n2, ...])
A = randn(..., type)
randn('seed', s);
Description
randn builds a scalar pseudo-random number chosen from a normal distribution with zero mean and unit variance. The default generator is based on a scalar 64-bit seed, which theoretically has a period of 2^64-2^32 numbers. This seed can be set with the arguments randn('seed',s), where s is a scalar. The seed is the same as the seed of rand and rng. randn('seed',s) returns the empty array [] as output argument. To discard it, the statement should be followed by a semicolon. The generator can be changed with rng.
randn(n), randn(n1,n2,...) and randn([n1,n2,...]) return an n-by-n square array or an array of arbitrary size whose elements are pseudo-random numbers chosen from a normal distribution.
An additional input argument can be used to specify the type of the result. It must be the string 'double' (default) or 'single'.
Examples
randn
1.5927
randn(1, 3)
0.7856 0.6489 -0.8141
randn('seed',0);
randn
1.5927
See also
repmat
Replicate an array.
Syntax
B = repmat(A, n) B = repmat(A, m, n) B = repmat(A, [n1,...])
Description
repmat creates an array with multiple copies of its first argument. It can be seen as an extended version of ones, where 1 is replaced by an arbitrary array.
With 3 input arguments, repmat(A,m,n) replicates array A m times vertically and n times horizontally. The type of the first argument (number, character, logical, cell, or structure array) is preserved.
With two input arguments, repmat(A,n) produces the same result as repmat(A,n,n).
With a vector as second argument, the array can be replicated along more than two dimensions; repmat(A,m,n) produces the same result as repmat(A,[m,n]).
Examples
repmat([1,2;3,4], 1, 2)
1 2 1 2
3 4 3 4
repmat('abc', 3)
abcabcabc
abcabcabc
abcabcabc
See also
zeros, ones, operator :, kron, replist
reshape
Rearrange the elements of an array to change its shape.
Syntax
A2 = reshape(A1) A2 = reshape(A1, n1, n2, ...) A2 = reshape(A1, [n1, n2, ...])
Description
reshape(A1) gives a column vector with all the elements of array A1. If A1 is a variable, reshape(A1) is the same as A1(:).
reshape(A1,n1,n2,...) or reshape(A1,[n1,n2,...]) changes the dimensions of array A1 so that the result has m rows and n columns. A1 must have n1*n2*... elements; read row-wise, both A1 and the result have the same elements.
When dimensions are given as separate elements, one of them can be replaced with the empty array []; it is replaced by the value such that the number of elements of the result matches the size of input array.
Remark: code should not rely on the internal data layout. Array elements are currently stored row-wise, but this may change in the future. reshape will remain consistant with indexing, though; reshape(A,s)(i)==A(i) for any compatible size s.
Example
reshape([1,2,3;10,20,30], 3, 2) 1 2 3 10 20 30 reshape(1:12, 3, []) 1 2 3 4 5 6 7 8 9 10 11 12
See also
rng
State of random number generator.
Syntax
rng(type) rng(seed) rng(seed, type) rng(state) state = rng
Description
Random (actually pseudo-random) number generators produce sequences of numbers whose statistics make them difficult to distinguish from true random numbers. They are used by functions rand, randi, randn and random. They are characterized by a type string and a state.
With a numeric input argument, rng(seed) sets the state based on a seed. The state is usually an array of unsigned 32-bit integer numbers. rng uses the seed to produce an internal state which is valid for the type of random number generator. The default seed is 0.
With a string input argument, rng(type) sets the type of the random number generator and resets the state to its initial value (default seed). The following types are recognized:
- 'original'
- Original generator used until LME 6.
- 'mcg16807'
- Multiplicative congruential generator. The state is defined by s(i+1)=mod(a*s(i),m) with a=7^5 and m=2^31-1, and the generated value is s(i)/m.
- 'mwc'
- Concatenation of two 16-bit multiply-with-carry generators. The period is about 2^60.
- 'kiss' or 'default'
- Combination of mwc, a 3-shift register, and a congruential generator. The period is about 2^123.
With two input arguments, rng(seed,type) sets both the seed and the type of the random number generator.
With an output argument, state=rng gets the current state, which can be restored later by calling rng(state). The state is a structure.
Examples
rng(123);
R = rand(1,2)
R =
0.2838 0.4196
s = rng
s =
type: 'original'
state: real 2x1
rawmax: 4294967296
R = rand
R =
0.5788
rng(s)
R = rand
R =
0.5788
Reference
The MWC and KISS generators are described in George Marsaglia, Random numbers for C: The END?, Usenet, sci.stat.math, 20 Jan 1999.
See also
rot90
Array rotation.
Syntax
A2 = rot90(A1) A2 = rot90(A1, k)
Description
rot90(A1) rotates array A1 90 degrees counter-clockwise; the top left element of A1 becomes the bottom left element of A2. If A1 is an array with more than two dimensions, each plane corresponding to the first two dimensions is rotated.
In rot90(A1,k), the second argument is the number of times the array is rotated 90 degrees counter-clockwise. With k = 2, the array is rotated by 180 degrees; with k = 3 or k = -1, the array is rotated by 90 degrees clockwise.
Examples
rot90([1,2,3;4,5,6]) 3 6 2 5 1 4 rot90([1,2,3;4,5,6], -1) 4 1 5 2 6 3 rot90([1,2,3;4,5,6], -1) 6 5 4 3 2 1 fliplr(flipud([1,2,3;4,5,6])) 6 5 4 3 2 1
See also
setdiff
Set difference.
Syntax
c = setdiff(a, b) (c, ia) = setdiff(a, b)
Description
setdiff(a,b) gives the difference between sets a and b, i.e. the set of members of set a which do not belong to b. Sets are any type of numeric, character or logical arrays, or lists or cell arrays of character strings. Multiple elements of input arguments are considered as single members; the result is always sorted and has unique elements.
The optional second output argument is a vector of indices such that if (c,ia)=setdiff(a,b), then c is a(ia).
Example
a = {'a','bc','bbb','de'};
b = {'z','bc','aa','bbb'};
(c, ia) = setdiff(a, b)
c =
{'a','de'}
ia =
1 4
a(ia)
{'a','de'}
See also
unique, union, intersect, setxor, ismember
setxor
Set exclusive or.
Syntax
c = setxor(a, b) (c, ia, ib) = setxor(a, b)
Description
setxor(a,b) performs an exclusive or operation between sets a and b, i.e. it gives the set of members of sets a and b which are not members of the intersection of a and b. Sets are any type of numeric, character or logical arrays, or lists or cell arrays of character strings. Multiple elements of input arguments are considered as single members; the result is always sorted and has unique elements.
The optional second and third output arguments are vectors of indices such that if (c,ia,ib)=setxor(a,b), then c is the union of a(ia) and b(ib).
Example
a = {'a','bc','bbb','de'};
b = {'z','bc','aa','bbb'};
(c, ia, ib) = setxor(a, b)
c =
{'a','aa','de','z'}
ia =
1 4
ib =
3 1
union(a(ia),b(ib))
{'a','aa','de','z'}
Set exclusive or can also be computed as the union of a and b minus the intersection of a and b:
setdiff(union(a, b), intersect(a, b))
{'a','aa','de','z'}
See also
unique, union, intersect, setdiff, ismember
size
Size of an array.
Syntax
v = size(A) (m, n) = size(A) m = size(A, i)
Description
size(A) returns the number of rows and the number of elements along each dimension of array A, either in a row vector or as scalars if there are two output arguments or more.
size(A,i) gives the number of elements in array A along dimension i: size(A,1) gives the number of rows and size(A,2) the number of columns.
Examples
M = ones(3, 5);
size(M)
3 5
(m, n) = size(M)
m =
3
n =
5
size(M, 1)
3
size(M, 2)
5
See also
sort
Array sort.
Syntax
(A_sorted, ix) = sort(A) (A_sorted, ix) = sort(A, dim) (A_sorted, ix) = sort(A, dir) (A_sorted, ix) = sort(A, dim, dir) (list_sorted, ix) = sort(list) (list_sorted, ix) = sort(list, dir)
Description
sort(A) sorts separately the elements of each column of array A, or the elements of A if it is a row vector. The result has the same size as A. Elements are sorted in ascending order, with NaNs at the end. For complex arrays, numbers are sorted by magnitude.
The optional second output argument gives the permutation array which transforms A into the sorted array. It can be used to reorder elements in another array or to sort the rows of a matrix with respect to one of its columns, as shown in the last example below. Order of consecutive identical elements is preserved.
If a second numeric argument dim is provided, the sort is performed along dimension dim (columns if dim is 1, rows if 2, etc.)
An additional argument can specify the ordering direction. It must be the string 'ascending' (or 'a') for ascending order, or 'descending' (or 'd') for descending order. In both cases, NaNs are moved to the end.
sort(list) sorts the elements of a list, which must be strings. Cell arrays are sorted like lists, not column-wise like numeric arrays. The second output argument is a row vector. The direction can be specified with a second input argument.
Examples
sort([3,6,2,3,9,1,2])
1 2 2 3 3 6 9
sort([2,5,3;nan,4,2;6,1,1])
2 1 1
6 4 2
nan 5 3
sort([2,5,3;nan,4,2;6,1,1], 'd')
6 5 3
2 4 2
nan 1 1
sort({'def', 'abcd', 'abc'})
{'abc', 'abcd', 'def'}
To sort the rows of an array after the first column, one can obtain the permutation vector by sorting the first column, and use it as subscripts on the array rows:
M = [2,4; 5,1; 3,9; 4,0] 2 4 5 1 3 9 4 0 (Ms, ix) = sort(M(:,1)); M(ix,:) 2 4 3 9 4 0 5 1
Algorithm
Shell sort.
See also
squeeze
Suppression of singleton dimensions of an array.
Syntax
B = squeeze(A)
Description
squeeze(A) returns an array with the same elements as A, but where dimensions equal to 1 are removed. The result has at least 2 dimensions; row and column vectors keep their dimensions.
Examples
size(squeeze(rand(1,2,3,1,4))) 2 3 4 size(squeeze(1:5)) 1 5
See also
sub2ind
Conversion from row/column subscripts to single index.
Syntax
ind = sub2ind(size, i, j) ind = sub2ind(size, i, j, k, ...)
Description
sub2ind(size,i,j) gives the single index which can be used to retrieve the element corresponding to the i:th row and the j:th column of an array whose size is specified by size. size must be either a scalar for square matrices or a vector of two elements or more for other arrays. If i and j are arrays, they must have the same size: the result is calculated separately for each element and has the same size.
sub2ind also accepts sizes and subscripts for arrays with more than 2 dimensions. The number of indices must match the length of size.
Example
M = [3, 6; 8, 9]; M(2, 1) 8 sub2ind(size(M), 2, 1) 7 M(3) 8
See also
tril
Extraction of the lower triangular part of a matrix.
Syntax
L = tril(M) L = tril(M,k)
Description
tril(M) extracts the lower triangular part of a matrix; the result is a matrix of the same size where all the elements above the main diagonal are set to zero. A second argument can be used to specify another diagonal: 0 is the main diagonal, positive values are above and negative values below.
Examples
M = magic(3) M = 8 1 6 3 5 7 4 9 2 tril(M) 8 0 0 3 5 0 4 9 2 tril(M,1) 8 1 0 3 5 7 4 9 2
See also
triu
Extraction of the upper triangular part of a matrix.
Syntax
U = triu(M) U = triu(M,k)
Description
tril(M) extracts the upper triangular part of a matrix; the result is a matrix of the same size where all the elements below the main diagonal are set to zero. A second argument can be used to specify another diagonal; 0 is the main diagonal, positive values are above and negative values below.
Examples
M = magic(3) M = 8 1 6 3 5 7 4 9 2 triu(M) 8 1 6 0 5 7 0 0 2 triu(M,1) 0 1 6 0 0 7 0 0 0
See also
union
Set union.
Syntax
c = union(a, b) (c, ia, ib) = union(a, b)
Description
union(a,b) gives the union of sets a and b, i.e. it gives the set of members of sets a or b or both. Sets are any type of numeric, character or logical arrays, or lists or cell arrays of character strings. Multiple elements of input arguments are considered as single members; the result is always sorted and has unique elements.
The optional second and third output arguments are vectors of indices such that if (c,ia,ib)=union(a,b), then elements of c are the elements of a(ia) or b(ib); the intersection of a(ia) and b(ib) is empty.
Example
a = {'a','bc','bbb','de'};
b = {'z','bc','aa','bbb'};
(c, ia, ib) = union(a, b)
c =
{'a','aa','bbb','bc','de','z'}
ia =
1 3 2 4
ib =
3 1
a(ia)
{'a','bbb','bc','de'}
b(ib)
{'aa','z'}
Set exclusive or can also be computed as the union of a and b minus the intersection of a and b:
setdiff(union(a, b), intersect(a, b))
{'a','aa','de','z'}
See also
unique, intersect, setdiff, setxor, ismember
unique
Keep unique elements.
Syntax
v2 = unique(v1) list2 = unique(list1) (b, ia, ib) = unique(a)
Description
With an array argument, unique(v1) sorts its elements and removes duplicate elements. Unless v1 is a row vector, v1 is considered as a column vector.
With an argument which is a list of strings, unique(list) sorts its elements and removes duplicate elements.
The optional second output argument is set to a vector of indices such that if (b,ia)=unique(a), then b is a(ia).
The optional third output argument is set to a vector of indices such that if (b,ia,ib)=unique(a), then a is b(ib).
Examples
(b,ia,ib) = unique([4,7,3,8,7,1,3])
b =
1 3 4 7 8
ia =
6 3 1 2 4
ib =
3 4 2 5 4 1 2
unique({'def', 'ab', 'def', 'abc'})
{'ab', 'abc', 'def'}
See also
sort, union, intersect, setdiff, setxor, ismember
unwrap
Unwrap angle sequence.
Syntax
a2 = unwrap(a1) a2 = unwrap(a1, tol) A2 = unwrap(A1, tol, dim)
Description
unwrap(a1), where a1 is a vector of angles in radians,
returns a vector a2 of the same length, with the same values modulo
With two input arguments, unwrap(a1,tol) reduces the difference between
two consecutive values only if it is larger (in absolute value) than tol.
If tol is smaller than
With three input arguments, unwrap(A1,tol,dim) operates along dimension dim. The result is an array of the same size as A1. The default dimension for arrays is 1.
Example
unwrap([0, 2, 4, 6, 0, 2]) 0.00 2.00 4.00 6.00 6.28 8.28
See also
zeros
Zero array.
Syntax
A = zeros(n) A = zeros(n1,n2,...) A = zeros([n1,n2,...]) A = zeros(..., type)
Description
zeros builds an array whose elements are 0. The size of the array is specified by one integer for a square matrix, or several integers (either as separate arguments or in a vector) for an array of any size.
An additional input argument can be used to specify the type of the result. It must be the string 'double', 'single', 'int8', 'int16', 'int32', 'int64', 'uint8', 'uint16', 'uint32', or 'uint64' (64-bit arrays are not supported on all platforms).
Examples
zeros([2,3])
0 0 0
0 0 0
zeros(2)
0 0
0 0
zeros(1, 5, 'uint16')
1x5 uint16 array
0 0 0 0 0